Optimal. Leaf size=1549 \[ \text {result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.61, antiderivative size = 1549, normalized size of antiderivative = 1.00, number of steps
used = 100, number of rules used = 30, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.500, Rules used =
{6025, 6024, 2498, 269, 213, 2500, 2526, 2512, 266, 2463, 2441, 2352, 2437, 2338, 2440,
2438, 209, 2636, 12, 2520, 6820, 6139, 6031, 6057, 2449, 2497, 210, 5048, 4940, 4966}
\begin {gather*} x a^2+2 b \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) a-2 b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) a-b x \log \left (1-\frac {c}{x^2}\right ) a+b x \log \left (\frac {c}{x^2}+1\right ) a+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (\frac {c}{x^2}+1\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (x+\sqrt {-c}\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (x+\sqrt {c}\right )-b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {c}{x^2}+1\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {c}{x^2}+1\right ) \log \left (x+\sqrt {-c}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (x+\sqrt {-c}\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (x+\sqrt {-c}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x+\sqrt {-c}}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (x+\sqrt {c}\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (x+\sqrt {c}\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (x+\sqrt {c}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x+\sqrt {c}}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {x+\sqrt {c}}{2 \sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt {-c}}\right )\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (\frac {x}{\sqrt {-c}}+1\right )-\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {c-\sqrt {-c} x}{2 c}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {1}{2}-\frac {x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}+1\right )+b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 210
Rule 213
Rule 266
Rule 269
Rule 2338
Rule 2352
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2449
Rule 2463
Rule 2497
Rule 2498
Rule 2500
Rule 2512
Rule 2520
Rule 2526
Rule 2636
Rule 4940
Rule 4966
Rule 5048
Rule 6024
Rule 6025
Rule 6031
Rule 6057
Rule 6139
Rule 6820
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx &=\int \left (a^2-a b \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 \log ^2\left (1-\frac {c}{x^2}\right )+a b \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 \log ^2\left (1+\frac {c}{x^2}\right )\right ) \, dx\\ &=a^2 x-(a b) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+(a b) \int \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{4} b^2 \int \log ^2\left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{4} b^2 \int \log ^2\left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{2} b^2 \int \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx\\ &=a^2 x-a b x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} b^2 \int \frac {2 c \log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx+(2 a b c) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+(2 a b c) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx\\ &=a^2 x-a b x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )+(2 a b c) \int \frac {1}{-c+x^2} \, dx+(2 a b c) \int \frac {1}{c+x^2} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx-\left (b^2 c\right ) \int \left (-\frac {\log \left (1-\frac {c}{x^2}\right )}{2 \sqrt {c} \left (\sqrt {c}-x\right )}-\frac {\log \left (1-\frac {c}{x^2}\right )}{2 \sqrt {c} \left (\sqrt {c}+x\right )}\right ) \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx+\left (b^2 c\right ) \int \left (\frac {\sqrt {-c} \log \left (1+\frac {c}{x^2}\right )}{2 c \left (\sqrt {-c}-x\right )}+\frac {\sqrt {-c} \log \left (1+\frac {c}{x^2}\right )}{2 c \left (\sqrt {-c}+x\right )}\right ) \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\sqrt {-c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{\sqrt {c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{\sqrt {c}+x} \, dx+\left (2 b^2 c^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx-\left (2 b^2 c^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+\left (b^2 (-c)^{3/2}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx-\left (b^2 (-c)^{3/2}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx+\left (b^2 c^{3/2}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\left (b^2 c^{3/2}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx+\left (2 b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+\left (b^2 (-c)^{3/2}\right ) \int \left (\frac {\log \left (\sqrt {-c}-x\right )}{c x}-\frac {x \log \left (\sqrt {-c}-x\right )}{c \left (c+x^2\right )}\right ) \, dx-\left (b^2 (-c)^{3/2}\right ) \int \left (\frac {\log \left (\sqrt {-c}+x\right )}{c x}-\frac {x \log \left (\sqrt {-c}+x\right )}{c \left (c+x^2\right )}\right ) \, dx+\left (b^2 c^{3/2}\right ) \int \left (-\frac {\log \left (\sqrt {c}-x\right )}{c x}-\frac {x \log \left (\sqrt {c}-x\right )}{c \left (c-x^2\right )}\right ) \, dx-\left (b^2 c^{3/2}\right ) \int \left (-\frac {\log \left (\sqrt {c}+x\right )}{c x}-\frac {x \log \left (\sqrt {c}+x\right )}{c \left (c-x^2\right )}\right ) \, dx+\left (2 b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{x} \, dx+\left (b^2 \sqrt {-c}\right ) \int \frac {x \log \left (\sqrt {-c}-x\right )}{c+x^2} \, dx+\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{x} \, dx-\left (b^2 \sqrt {-c}\right ) \int \frac {x \log \left (\sqrt {-c}+x\right )}{c+x^2} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{x} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {x \log \left (\sqrt {c}-x\right )}{c-x^2} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{x} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {x \log \left (\sqrt {c}+x\right )}{c-x^2} \, dx+\left (2 b^2 c^{3/2}\right ) \int \left (-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c-x^2\right )}\right ) \, dx-\left (2 b^2 c^{3/2}\right ) \int \left (\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c+x^2\right )}\right ) \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+\left (b^2 \sqrt {-c}\right ) \int \left (-\frac {\log \left (\sqrt {-c}-x\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\log \left (\sqrt {-c}-x\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx-\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (-\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c}+x} \, dx-\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c}-x} \, dx-\left (b^2 \sqrt {-c}\right ) \int \left (-\frac {\log \left (\sqrt {-c}+x\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\log \left (\sqrt {-c}+x\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx-\left (b^2 \sqrt {c}\right ) \int \left (\frac {\log \left (\sqrt {c}-x\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\log \left (\sqrt {c}-x\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (-\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\left (b^2 \sqrt {c}\right ) \int \left (\frac {\log \left (\sqrt {c}+x\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\log \left (\sqrt {c}+x\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx-\left (2 b^2 \sqrt {c}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx-\left (2 b^2 \sqrt {c}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c-x^2} \, dx-\left (2 b^2 \sqrt {c}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\left (2 b^2 \sqrt {c}\right ) \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c+x^2} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{\sqrt {-c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{\sqrt {-c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{\sqrt {-c}-x} \, dx-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{\sqrt {-c}+x} \, dx-\left (i b^2 \sqrt {c}\right ) \int \frac {\log \left (1-\frac {i x}{\sqrt {c}}\right )}{x} \, dx+\left (i b^2 \sqrt {c}\right ) \int \frac {\log \left (1+\frac {i x}{\sqrt {c}}\right )}{x} \, dx-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{\sqrt {c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{\sqrt {c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{\sqrt {c}-x} \, dx-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{\sqrt {c}+x} \, dx-\left (2 b^2 \sqrt {c}\right ) \int \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx+\left (2 b^2 \sqrt {c}\right ) \int \left (-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (-\frac {-\sqrt {-c}-x}{2 \sqrt {-c}}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right )}{\sqrt {-c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {-c}-x\right )-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {-c}+x\right )+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (-\frac {-\sqrt {c}-x}{2 \sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right )}{\sqrt {c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {c}-x\right )-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {c}+x\right )-\left (b^2 \sqrt {c}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}-x} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}+x} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )+2 \left (b^2 \int \frac {\log \left (\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx\right )-b^2 \int \frac {\log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx-b^2 \int \frac {\log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx+2 \left (b^2 \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx\right )-b^2 \int \frac {\log \left (\frac {2 \left (\sqrt {-c}-x\right )}{\left (-1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx-b^2 \int \frac {\log \left (\frac {2 \left (\sqrt {-c}+x\right )}{\left (1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {-c}}\right )}{x} \, dx,x,\sqrt {-c}-x\right )+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {-c}}\right )}{x} \, dx,x,\sqrt {-c}+x\right )-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {c}}\right )}{x} \, dx,x,\sqrt {c}-x\right )+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {c}}\right )}{x} \, dx,x,\sqrt {c}+x\right )\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt {-c}}\right )\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {c-\sqrt {-c} x}{2 c}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {1}{2}-\frac {x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+2 \left (\left (i b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i x}{\sqrt {c}}}\right )\right )+2 \left (\left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {c}}}\right )\right )\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt {-c}}\right )\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {c-\sqrt {-c} x}{2 c}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {1}{2}-\frac {x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )\\ \end {align*}
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Mathematica [A]
time = 2.76, size = 565, normalized size = 0.36 \begin {gather*} a^2 x-2 a b \sqrt {\frac {c}{x^2}} x \left (\text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )+\tanh ^{-1}\left (\sqrt {\frac {c}{x^2}}\right )\right )+2 a b x \tanh ^{-1}\left (\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {\frac {c}{x^2}} x \left (-2 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )^2+4 \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right ) \tanh ^{-1}\left (\frac {c}{x^2}\right )-\frac {2 \tanh ^{-1}\left (\frac {c}{x^2}\right )^2}{\sqrt {\frac {c}{x^2}}}+2 \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right ) \log \left (1+e^{4 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )}\right )-2 \tanh ^{-1}\left (\frac {c}{x^2}\right ) \log \left (1-\sqrt {\frac {c}{x^2}}\right )+\log (2) \log \left (1-\sqrt {\frac {c}{x^2}}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {\frac {c}{x^2}}\right )\right )+2 \tanh ^{-1}\left (\frac {c}{x^2}\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log (2) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {\frac {c}{x^2}}\right )\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{4 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )}\right )-\text {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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